The Laplace Transform is an object of fundamental importance in pure and applied mathematics. In addition, it has special pedagogical value in that it can provide a natural and concrete setting for a student to begin thinking about the modern concepts of "operator" and "functional". Most undergraduate textbooks, however, merely define the transform, and go on to properties and applications. The present article is intended to help students discover the Laplace Transform for themselves. The emphasis, then, is neither elaboration of technique nor axiomatic proof, but initial discovery of the basic idea.